Question

The sum of inradius and circumradius of incircle and circumcircle of a regular polygon of side $n$ is

• A. $\frac{a}{4}\cot \frac{\pi }{2n}$
• B. $a\cot \frac{\pi }{n}$
• C. $\frac{a}{2}\cot \frac{\pi }{2n}$
• D. $a\cot \frac{\pi }{2n}$

Answer 1

Answer: (C)

$\frac{a}{2}\cot \frac{\pi }{2n}$

$r+R=\frac{a}{2}\cot \frac{\pi }{n}+\frac{a}{2}cosec\frac{\pi }{n}$
$=\frac{a}{2}\left(\frac{1+\cos \frac{\pi }{n}}{\sin \frac{\pi }{n}}\right)=\frac{a}{2}\frac{2{\cos }^2\frac{\pi }{2n}}{2\sin \frac{\pi }{2n}\cdot \cos \frac{\pi }{2n}}$
$=\frac{a}{2}\cot \frac{\pi }{2n}$.